Littlewood-Paley Estimates for Lévy Processes

$$L^p$$ inequalities for certain Littlewood-Paley functionals arising from Lévy processes will be discussed. These are motivated by applications to the $$L^p$$ boundedness of Fourier multiples which give $$L^p$$ regularity of solutions to some non-local operators, including the fractional Laplacian. Non-local operates have been extensively studied in recent years by researchers in analysis, probability and PDE. The relevant Fourier multiples have been studied using the deep sharp martingale transform inequalities of Burkholder. The proofs here, although not sharp, are completely elementary and use nothing more than Itô's formula and arguments similar to those for the classical case of the Laplacian.